Lecture07Print

Lecture07Print - © ¡¢ f g b T h f p i q U Ÿ  › š...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: © ¡¢ f g b T h f p i ¡ q U Ÿ  › š ™ ˜ — ” p – œ “ “ ž •– ’“ ’ ž • © ¡¢ T S £ £ V U TW a XY` ¤¥ b U cd de ¦§ © ¨     ¦ ¥ → ‰ ˆ ¨ ≠ ¦ Š ©  ¥ § $ ‹ ¦ ‰ © Ž    ‹   ˆŒ   !  Ž Š   ‘  ‘  ‡      &' % F F ' G F )0 ( "#$ A & G &' ( 2 1 34 H E 2 5 & H 6 5 G ' 9 & 0 G 5 P BI 78 C 9 9 ( H A 5@ 6 ' I A 2 B 5C )0 & A 0 & D 2 ' QR )0 E & © ¡¢ fg b T h ¥ © f ¦ © p i q U ¤ © p § © ¨ © ª © ¡¢ fg b £ £ T h f p i r r € y v x w q U p t t s s t t u u v w p z s { e m ™def t f d l dg h h p p w v z hi r  d }g r x j x x e y g Θ l  } s e m f h w p g ¤ £ § £ jk l r d i n v v ¥ £ w ¨ £ dp † ƒ w t g ™d g l h x o Θ y l p d q w n jk m ™dv t i d h h e ™de r ‚ g j i n h r n { e d ™l h l i w w t en r| g m e ™d m g en l o ™d h h d w e l ‚ ‚ ‚ƒ g ™d v x t w dp ™ h j „ j ™dv ™d q m f { t † … d g q e h g ‚‡ ™r v ™ d w i „… g j l l x p i ™ ™ e ˆ‰ l g r g n t } x ef u ™ jk jk ‘ e l l m ‘ d h l l } ≤ e h en d h e e es ™d i ˜ ˜ — ~ z { h g g x l ’“” ’“” l x i e w n o d l ¦ £ •– •– j t ™ € t t o ¢ c © ¡¢ f g b T h f p i q U p © ¡¢ f g b £ £ T h f p i œ q U ­ ¬ – “ p ž • ® ’ ¯ α º ¼ ¹ ¸ · ¶ µ ´ ³ » #       Ž # Ž °  ” ” š › ¡  — ˜ ” ˜ Ž # —   °  š #  Θ  Ž  Θ α º ¼ ¹ ¸ · ¶ µ ´ ³ »    α     Ž  ²  #    Ž « Ô © ¡¢ ¡¢ fg = b £ ô ó ïðñò ³ ³ £ T h f p i q U = Ð Ï Õ Ó È Í Ì è · ¸ =+ − ç p ¹ ¼ µ ´ α Î ¿ ‹ ³ µ + ¹ · Ñ ´ À ô ó  ïõö − ¹ Ó Ò Ê É Ë î 5 éêëì í =+ Á ‹ ÷ α − ç Ü Ú Û =+ ⋅ = Ø × Ö  à $  Ä Å ˆ −   ‹ Ú Ù ‹ ‹ =+  ‹ ⋅ α Θ ãä = á ß â å à − Þ Θ Ý Ã α  Ä =+ Ç ⋅ Æ  æ å  − à  å ¾ ˆ $ Ä Å ¾ ¾ ¾ ¾ ¾ ½ ± ÿ þ ý ü û ú ¿ © ¡¢ ¡¢ ¢ ¡ f g b £ £ T h f p i ¬ – “ “ “ œ “ “ § œ œ ¦ œ q U ¨ – ® ’ ¬ ¬ ¬ ’ ’ ’ – – ® p § • • • £ © § ¥ ¤   Ž #  ¥ ¤ °      ©   ©  © –  œ    © © © k  f © ¨ © ¨ © ¨  p o #  œ “    h v  j ¬  œ  Ž w { dp £ α £ £ ù £ ¾ ¾ ¾ ¾ e e ø © ¡¢ fg b T h f p i q U p Œ § © à ¦    ©   Ä Ã  ¡¢ £ £ ≤ ÿ þ ý ü û ú ¿ ¢ ¡ # ## # # “ œ        °     ’ Ž Ž Ž Ž Ž Ž #    ° –              ‰ ˆ Š Ž #  ¥ °   $ ‹  ‰  ‹   ˆŒ #    ˆ    Ž e e d c © ¡¢ f g b T h f p i q U p $   † ¥ © ¡¢ " f g b = £ £ T h ∞ = f p i   q U ! α p  α −α    α −   = †  −α  α     α   α e e « ‡ © fg b $ ‹ T h f ‰ p i q U p ¥ $ ‹ ˆ ‰ ‰ ‰ ‰  ¦ ‹ ' % % % % ( Š Š Š  © ¡¢ ¡¢ fg b £ £ T h f p i  q U p % ⋅ ⋅  #  = "  Š Š ( Š ˆŒ Š Š Š  Š Š Š Ž  $  À −  ⋅ ( Š ‰ & Š Š " "    Ž  & ⋅   ° # Š ( Š Š Š "   Ž  ˆ Œ  $  & ) & ⋅    # " Ž ° = ⋅ ( Š ‰ ˆ  ° Š  Š ‘Š Š # Š ‹ ˆŒ ‰  $   # & " Ž ° e e ± ¢ ¥A © ¡¢ f g b T h f p i q U p A Ä — à ¨ ¥ ¦ C ¥ Ä  © Œ ¥ ‘ à C  ¨ ¥ ¨  ¥ ¦ ¥  ‰ §  ‰  ¦  ‰ § à ¥      Š ¥ ‰ $ ‰   ‰ § ’“ ¥” § Æ $ Æ •  ‰ § Å ‰ Œ –  ‰ Å © ¡¢ f g 012 092 2 01 b £ B £ T h ¦ 3 @ f 3 p i C  3 @ @ q U 6 45 C 6 5 4 6 45 p  $ 01 ¦ 01 01 D 1 1 1 ‰ “ ¯ “ ¯ “ ¯ “ Š “ “ ¯ ¯ ¯ ¯ ¯ ¯ ¯ “ ¯ ¯ ¯“ ¯“ 5 ¯ 5 5 ¥ £78 £78 £ 78 $  1 £ 1 1 ¯“ “ ¯ ¯“ ≤ “  ¯ Ž ¯ Ž “ ¯ ¯ 5 6 5 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ 6 ¯ 5 ¯ “ “ ¯ ¯ ¯ ¯ ¯ ¯“ “ ¯ 5 ¯5 6 5 6 5 1 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯“ ¯ “ 6 5 6 ¯ 5 ¬ ¯ ) ) e e ø ² ” © ¡¢ fg b T h f p i q U p  † †  ù    † †    † © ¡¢ 0 fg ” b 3 0 £ £ Ÿ y T a h V 3 ‡ 01 f a ˜ — S p i s 01 q U Ÿ ˜ 3 x V p 6 3 5 6 3 5 U ` ` T W i R V d ª X d ª G W I Q ` f V e ¯ Y 0 S F Q r ª ‚ 3 I a W c £ 78 HIP 01 X V t E G U `R ` 3 5 Q œ Q R • e R ` 6 T f V q 3 I T S W g ` £78 Y X TW UV R P Q † V V E X W ` 6 ¯ W  W r f X V 3 T S H ¯ H R d ª ªd V “ ¯ R `x x V q 3 f e I e ¯ W € d ª h ¯ € “ 0 “ ¯ P Q – “ “ c c 3 X R S “ X X a V V E W “ R E E R ž e R  01u ’ g V Q U ’ I E £ 78 £78 HIP §v b a T S ` x £78 P V E 6 5 1 ` Q HIP P œ QR i R X π • “ ¯ ˆ b 6 Q R P “ f W I h V QR X S T w a ˆ I h ¯ R V ¯ F – V T S I ` I £78 ¯ q a W V I ¯ P π W § I “ ¯ W RUV ¨ E ’ ¯ ¨ R T ’ V “ ¯ 5 6 ¯ `R RY TW w 3 RY I V V g I G “ ¯ W G ` S ` G ` “ HI G ` XR E X “ ¯ V 0 f I W X V t R ¯ “ W X W I 3 £78 gs R E E “ t a £78 X IP S I F RY 0 1 R U tR U E V G “ ¯ ` F EF G … G ‚ ` F Q 3 V “ UV „ ` 3 5 “ ¯ t ƒ P 6 € T S ¤ 3 s ` “ R‚ W G p ¯ W Y i R R P Q £78 V ) G W E I X V ¯ V “ ` T F G V Qx “ £7 8 F ¯ T S X P G I X I s G EFG H I ¯ R ž ) R “ ¯ 6 X œ R E e “ c d ½ V f © ¡¢ f g V f b ‡ U ‚ V T h g ) R H W U ` ` 3 G G f P Q p i f ` X q U ‚ t R g p ` 6 Q  W G R I ˆ V Q ` R ‚ G X 1 FG S – ˆ G T ` f ` V ` ­ X x U c W ˜ ‡› X H V ‹ 0 G Q ¤ 6 c I ž W 1 W U G ¡¢ S £ £ R V X f ` V E V s X R I S x W W I f I X Q R ` HI P G H EF € V T  ‚ S r 01 R t V V G S XU G I T S F Q Q t 6 £ G 5 V V V I c I g W W F ` X g E UV W o G T ` V t R V r t I V P t U x U F 01 W V X v G G QR ⋅ £ V ` ¯ “ g r ` † I – – ¯ t R V S – 6 H ¬ R f ’  V T V € d R X V I R t ) G Y 5 I V £ 5 T V I V ` Q R V V F W R S tR V T ` G n V s I G S G I £ X F T S X U q 9 G ` X € HI P V S ` ` V R t V H `R V ˆ X V Q r E V F X F W U S ¯ R r I E V c G S h I ™ I S I I ˜ ⋅ – ˆ ˆ † “ e ¯“ … ˆ  R Q R V G R I S f U s S G ` F F ` I V 6 ¯  V T U Q I X T S – t UV UV £ G HI P x V ` ` o G V r t t S ˜ P X P R ` I I “s “ r S c c E Q T T W d ™ R I I V V r R Q  HI P ` F G I H X R I V ’ R ª V Q UV ‚ S X V P E S Q ⋅ “ HI P 5 QRY ˜7 I U – ¦ `R G ¬ ’ 6 f ­ “ ˆ ˆ ) R V H U  R ` ™ g V ` P E X G S  V QR X R E I V G X x ‚ S R G T ˆ G s ` b £ F I G S S t V X G s P U f X G G S r x f HIP S  Q h `T ` F ‚ r g ˆ W I V V ˜  UV  ` R S X P I V Q t Q t V V X P r E X I S 9 ` ` Q = P S I ˆ T ‡ £ E V ˜ — R G r P X X I X ˆ e £ G v  ™ r T S U V P V G ¯ ` R R ‚ S V  € V p X V Q T ` F E ˆ `  U  q G I G R ‘ = R E V S q UV Q g Q “ S š V S ` S t Q P ET ∈ I E r V f V HI E f X Q T V 6 r ˜ t X ` v G g S  a s ` RU ‹ ‰ “s R V x ∪ X r ¯ p E x X ˜ 3 UV V c S  Q I W Ž W – x P 9  G R T S UV `R X ` € d G g G ` Q E – § G I S ˜ ` ` r ¿ T • ” – ‚ S TW s R ` R ¦ ‹ 0 I s X @ q W ­ R 7 T ` Š Y t 6 `R V X S q £ Q I  ¨ ` V I V V y I € P E I S V I V I T g  G E I T X ž I ` r P V W W – g X © X I X U T S S t V X 1 Q HIP t F ` 3 G Q – `R I I HP ` S W ` Q HI ` F Y 8 ` w ‹ 0 T @ H ` F ¬ R  R UV ` V V ¨ t X TW S Œ c UV T V I Q t G P V QR o V § G G X I R Q G S t S V Q wx X Q P Q f i R V x “ ` R E F G ‚ € V 6 V Xs † Q I G T S W T V r 3 “ HG I f R I `R H W x ‚ V R € T X X V “ ¯ F ˜ 5  ) G ` ` G R E T x S e I G E S I V d t R Q c i X RUV def E € r QR F I P R ‚ W V G r S V € S w G X ` G F T W 6  ˆ b 3 ` I ‚ S t t V V y o ` F G UV U X X G V I g I £ F ˜ h o R UV 6 `T P t H f G ˆ ghi w hi R t T V V – t o X Q G P w r I ` F HI P ` T ` F ` R X ` G G X V S Y S R V UV R T ‡ 3 ‚ s S I T I ` I o G V jkl Q g ’ ¬ F e wx T S t “ tu m H P ` V h U x x G I Q I S R I X R W i Y rI X R ` W V ) T Q I R E R R V S S S R r ) `T W R ˜ W ) d s X V E F ` r V l ` W ` X m k ˜ I I V f R s r V UV X TW ` P Q I W j G G ) R R  Qx F X € x  t UV X X T G P I P F V ` R H R X G `R R s V v c UV Q T c x S U X P U H U r v  R X R t s X r R X E P ` s S Q t S Q Q e rI r c c S S S ‡ © ¡¢ fg T S f b 3 T h b £ c f c v p i © ¡¢ fg b £ £ T h f p i ≠ X f X 6 q U R E b q U E y p p R S f U H T S Y U X f I r “ G ` V ‹ 0 x S U H 1 V ` P I ` S 6 rT £ 5 W  G F ˜ f ` I 0 X R W UV S ) 3 ` g W ` S P ‹ § 0 ` `R S ª 1 r X R ² ™ S T v v œ  G  f ­ E I X R ¶ 5 ¥  V µ UV V G c ˜ ² F QR g P W ` V G S g r ` s ` ` S − g − x r ` Y R G ` W V  X x §  S ´ R HIP ª t t I V E ² Q œ F  Q ¬ I G U T V HIP Q T ¤ S ˜ ‡ Q V ª S R W U Xs R G S t G = X ≡− = ¸ · f c Q T r ‚ UV S r g S f r ` `R W g V ƒ = © ¨ ± ¹ E ` ¢  V S « b 3 E T € V Q r = ˜ ‡ g V c x i g s f 1 £ ` T I I ¸ ­ R E S HIP € V ©   U s x g µ¶ gR r z V ¥¦ ` S t R V “ ` ∈ ˜ ‡› ¸ T I HIP g © G 5 Q H  S −  € }~ 6 ¤ x X ` − ƒ x Q Q œ S g f { ´ Y c ¤ T t V r ˆ Q ≠ G E v g I ¸ Q HIP X ¬ X £ 6 ‚ ` T © v ˜ V “ E | x E I U £ x V i ≡ R I V HIP V g S ®¯ ­ Q I t V ` R ` F d ‚ g ¡ ` v s X W V ` ˜ T S UV I HP t x HI P Q ˆ b t T V Q P f  ˆ ˆ I R ¤ 3 žŸ  = ° h V Q V R ` £ V z T v ˆ ˜ ‡› V h V Q I T Q ˆ G HIP v W x S ¤ ³ U … T T V S x V S ¬ V ‡ 6 X T U r x G V € e I X W ‡ 3 o „ V S − I œ ) R X Q R V R P I S V ` E £ ` i T S T S rT t 9 S E ` V S R G S v F V QR V I 6 Y T  G R V o I G s V b 3 V G ‹ £ 0 G V X V I E E ) G UV i I T s U i V Q V X ` W ` V W I Q d I S T I s t H Q ` t U QI `T x £ UV R T  s S ˆ ` b £ F =α − T ` c R ` F UV R U i T I V x UV H X c UV G h R F V XR U r Y `R R V t ˆ G P V X t T E I I P „ V r I R E ‚ S U U R I  R G R T S R r ‡ £ W g t V V X V e o ` I P ‚ X S T I P ` rT T ` P ` V I P G E ` F X c X t o 1 G R I R `R U X V S R  F `R g T I U I  V  ˆ ` R R V I ¤ 3 W ` Q Q Q G Q T ‚ t I R ) R ‚ h E P ` ) U R U S ` ‚ ˆ V ) R ) U P T X c V ) P ` P U V H U H R r W x X t I X W R G ` H U r T t ‡ 3  t G r T R r I 5 €  QR R S Q R Q Q Q c c ¢ c © ¡¢ f g b T h f p i q U p © ¡¢ f g b £ £ T h f p i q U p s ` H ² ` ­ Ç Ã ¶ Î Í µ » ² Ï P ≡− = É È Â Ð É ¾ ÆÇ É − X  ´ ² Q f ¬ b 3 g ` i ` – S t G ≡− V s ` = ⋅ − ` H RY ` Å V ¸ · ± – ¹ 6 ≤ É Ë  Æ Ã  Å Ñ Ã Ò ¿À Ä ` H ½ ¸ ­ Ê Ì º “ P µ¶ ¾ X IP X ¸ − 1 f s R  − W I G ½ ˜ 7 ´ ` Q Y ¸ HV ¬ S ` V r ¬ ¦ − S  UV t ¾ º T S Á P ` V r ½ G X ®¯ ­ 5 X IP 6 V º R E Y V ° “ „ g b £ ⋅ ` ` “ 6 º t I ¼ ³ X I P ¬ ⋅ v T b Q UV 6 ≠ c ² ¡¢ “ » ` ` R G W R 1 ≠ W R V ¯ ˜ 7 V RUV T ¬ I U ¦ 5 QR R W W I R V V P T T S S ` G U Q Q r S TW G “ 1 R X W R V ˜7 V ˜ X TW T ¦ ˜ 5 Q rT S T ) RU X ‚ U r c E € r c « © fg b T h f p i q U p  © ¡¢ fg b £ £ T h f p i q U p α Ó  α Õ Ô Ó + † α Ó α  #  c c ½ ± © ¡¢ f g b T h f p i q U p £ + Ø Û + Þß Ø Ö ØÙÚ Ü Ý −α Ý Ý à −α × c ø ¡¢ £ ...
View Full Document

This note was uploaded on 03/12/2011 for the course CS 234218 taught by Professor Geiger during the Winter '01 term at Technion.

Ask a homework question - tutors are online